Large Time Behavior of Solutions to the Nonlinear Heat Equation with Absorption with Highly Singular Antisymmetric Initial Values

Abstract

In this paper, we study global well-posedness and long-time asymptotic behavior of solutions to the nonlinear heat equation with absorption, ut-Δ⁢u+|u|α⁢u=0{u_{t}-\Delta u+\lvert u\rvert^{\alpha}u=0}, where u=u⁢(t,x)∈ℝ{u=u(t,x)\in\mathbb{R}}, (t,x)∈(0,∞)×ℝN{(t,x)\in(0,\infty)\times\mathbb{R}^{N}} and α>0{\alpha>0}. We focus particularly on highly singular initial values which are antisymmetric with respect to the variables x1,x2,…,xm{x_{1},x_{2},\ldots,x_{m}} for some m∈{1,2,…,N}{m\in\{1,2,\ldots,N\}}, such as u0=(-1)m∂1∂2⋯∂m|⋅|-γ∈𝒮′(ℝN){u_{0}=(-1)^{m}\partial_{1}\partial_{2}\cdots\partial_{m}\lvert\,{\cdot}\,% \rvert^{-\gamma}\in\mathcal{S}^{\prime}(\mathbb{R}^{N})}, 00{\alpha>0}. Our approach is to study well-posedness and large time behavior on sectorial domains of the form Ωm={x∈ℝN:x1,…,xm>0}{\Omega_{m}=\{x\in\mathbb{R}^{N}:x_{1},\ldots,x_{m}>0\}}, and then to extend the results by reflection to solutions on ℝN{\mathbb{R}^{N}} which are antisymmetric. We show that the large time behavior depends on the relationship between α and 2γ+m{\frac{2}{\gamma+m}}, and we consider all three cases, α equal to, greater than, and less than 2γ+m{\frac{2}{\gamma+m}}. Our results include, among others, new examples of self-similar and asymptotically self-similar solutions.

Keywords

• heat equation
• absorption
• complexity
• sectorial domains
• asymptotic behavior
• antisymmetric solutions
• scaling
• self-similar solutions
• 35k55
• 35b40
• 35c06
• 35b30
• 47a20